Elementary Differential Geometry

Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout.

New features of this revised and expanded second edition include:

• a chapter on non-Euclidean geometry, a subject that is of great importance in the history of mathematics and crucial in many modern developments. The main results can be reached easily and quickly by making use of the results and techniques developed earlier in the book. Coverage of topics such as: parallel transport and its applications; map colouring; holonomy and Gaussian curvature.
• Around 200 additional exercises, and a full solutions manual for instructors, available via www.springer.com

Table of Contents

1. Curves in the plane and in space

2. How much does a curve curve?

3. Global properties of curves

3. Global properties of curves

4. Surfaces in three dimensions

5. Examples of surfaces

6. The first fundamental form

7. Curvature of surfaces

8. Gaussian, mean and principal curvatures

9. Geodesics

10. Gauss’ Theorema Egregium

11. Hyperbolic geometry

12. Minimal surfaces

13. The Gauss–Bonnet theorem

A0. Inner product spaces and self-adjoint linear maps

A1. Isometries of Euclidean spaces

A2. Mobius transformations